Mathematical Olympiad Rioplatense

A 3-digit number with equal digits is divisible by $111$ and hence by $37$ because $111=37\cdot 3$. By hypothesis such a number is $1+2+\cdots +(N-1)=\frac{1}{2}N(N-1)$, so $N(N-1)$ is divisible by $37$. Hence $N=37k$ or $N=37k+1$ for some integer $k\ge 1$. If $k\ge 2$ then $N\ge 74$, $\frac{1}{2}N(N-1)>1000$ and $\frac{1}{2}N(N-1)$ is not a 3-digit number. Therefore $k=1$, i.e. $N=37$ or $N=38$. Since $1+2+\cdots +36=666$, $1+\cdots +37=703$, only $N=37$ is a solution.